Want to add bonds to your Nitrogen portfolios? See Working with Bonds in Nitrogen Portfolios for a detailed walkthrough.

## Overview

Nitrogen models bonds based on maturity date, coupon rate, periodicity[1], bond type[2], and price, in conjunction with estimates of bond market movements should Advanced Risk Modeling be enabled. Nitrogen does not specifically incorporate other factors such as credit rating, optionality, or bond type by issuer (municipal/federal/corporate/etc.), though such factors are arguably included in the bond’s price post issue date.

Generally, bond duration is the weighted average time of payment of the cash flows of a bond, but for our purposes duration represents the price sensitivity of a bond to changes in a benchmark yield. Greater duration means a given change in interest rates will have a greater impact on a bond’s price. For example, a 3% coupon bond with a duration of 7 years will lose approximately 7% of its value if rates rise to 4%, and vice versa if rates drop.

To determine Duration we first determine the Yield to Maturity for the bond. For Coupon Bonds, this is achieved by using Newton’s method to recursively solve for it such the Dirty Price (Quote Price plus Accrued Interest) minus the Present Value of the Bond equals 0. This is done through at least five iterations with the initial guess being the bond’s coupon rate. From here it is quite simple to obtain the periodic discount factor to determine each cash flow’s present value, weight it by time, and determine the Bond’s Macaulay Duration. Once this is determined we then convert it to Modified Duration which will give us our first order approximation to the bond’s interest rate risk. Now, there are of course special circumstances with Zero-Coupon Bonds and Perpetual Bonds. For Zero-Coupon Bonds since there is only one payment its Macaulay Duration will equal its Maturity. With Perpetual Bonds its Macaulay Duration will equal one plus Yield to Maturity over Yield to Maturity. It is important to note that we do not assume any day count convention (i.e. we do not assume all years have 360 days, or 365 days, etc.). Rather, we use the actual number of days per year over the given time period for which the bond exists. Further, all Yield to Maturities use Street Convention.

Duration is a linear approximation and would suggest that, in the previous example above, rates rising to 5% would cause a price drop of approximately 14%. In fact, duration also decreases as rates rise, so we would see a reduced sensitivity to yield changes as rates increased. This brings us to Convexity. To review, Convexity is a measure of the non-linear relationship between a change in a bond’s price and a change in rate and is the second order factor. Convexity can be generally thought of as the rate of change of a price function's slope (its second derivative) and in this case the slope is the rate of change of a bond’s price relative to a change in rates. The more quickly the slope of the curve changes in one direction the more convex the curve is. In the case of bonds this means that the greater the convexity, the more rewarding the bond is to its owner (all else being equal). Namely, its value will increase more if interest rates drop and will decrease less if interest rates rise. Convexity is an asset to the bond holder.

For Coupon Bonds we calculate the convexity element of each cash flow where the Convexity of the Bond is equal to the sum of each individual convexity element of the bond divided by its Periodicity squared. Again, there are special circumstances with Zero-Coupon Bonds and Perpetuals. Once the Convexity of the bond has been determined the formula for estimating a bond’s price change ∆P due to a benchmark yield change ∆r with duration D and convexity C is ∆P/P = -D*∆r + C/2 * ∆r^2

Here is an example of this output represented by a bond’s Risk/Return Scenario in Nitrogen.

## Review

Using the previously mentioned bond analytics we can precisely determine the interest rate risk of each bond within our system to a six-month 95% confidence interval. Below is a brief and intuitive recap of some common types of bonds:

Further, here is a quick review of the effect of increased Duration on Municipal Bonds.

## Stress Tests

Nitrogen stress tests follow the same approach as above with respect to the price change of a bond but use different yield changes for each scenario. For example, the 2013 Bull Market saw the 10-year treasury go up 115bps.

## Retirement Maps

Retirement maps, when using a specific portfolio (rather than a Risk Number benchmark) for the underlying projection, will be affected by the same bond calculations that affect the portfolio. Note that a portfolio with bonds assumes 6-month return and volatility is applicable for the span of the retirement map, i.e. it assumes the portfolio could be proportionally reinvested into bonds with similar parameters (coupon rate, maturity) in perpetuity.

[1] If a specific Periodicity is not provided, we will assume the Coupon Bond pays semiannually and therefore has periodicity of 2.

[2] Nitrogen currently factors in the following bond types: Coupon Bonds; Zero-Coupon Bonds; and Perpetual Bonds. Callable and Puttable Bonds are assumed to be held until maturity.